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Directional bending wave propagation in periodically perforated plates

Andreassen, Erik; Manktelow, Kevin ; Ruzzene, Massimo

Published in:Journal of Sound and Vibration

Link to article, DOI:10.1016/j.jsv.2014.09.035

Publication date:2015

Link back to DTU Orbit

Citation (APA):Andreassen, E., Manktelow, K., & Ruzzene, M. (2015). Directional bending wave propagation in periodicallyperforated plates. Journal of Sound and Vibration, 335, 187–203. https://doi.org/10.1016/j.jsv.2014.09.035

Directional bending wave propagation in periodically

perforated plates

Erik Andreassena,∗, Kevin Manktelowb, Massimo Ruzzeneb

aDepartment of Mechanical Engineering, Technical University of Denmark, Nils KoppelsAlle, Building 404, Denmark

bSchool of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332

Abstract

We report on the investigation of wave propagation in a periodically perfo-

rated plate. A unit cell with double-C perforations is selected as a test arti-

cle suitable to investigate two-dimensional dispersion characteristics, group

velocities, and internal resonances. A numerical model, formulated using

Mindlin plate elements, is developed to predict relevant wave characteristics

such as dispersion, and group velocity variation as a function of frequency

and direction of propagation. Experimental tests are conducted through a

scanning laser vibrometer, which provides full wave field information. The

analysis of time domain wave field images allows the assessment of plate

dispersion, and the comparison with numerical predictions. The obtained

results show the predictive ability of the considered numerical approach and

illustrate how the considered plate configuration could be used as the basis

for the design of phononic waveguides with directional and internal resonant

characteristics.

Keywords: Perforated plate, Double-C resonator, Phononic plate,

∗E-mail: [email protected]

Preprint submitted to Journal of Sound and Vibration September 24, 2014

Scanning laser vibrometer.

1. Introduction

Recently, materials and structures with periodic modulations in their

physical properties have attracted significant attention due to their unusual

electromagnetic, acoustic and elastic properties. Such properties lead to

unique wave propagation characteristics that can be exploited for the design

of waveguides, filters, focusing devices, as well as structural assemblies with

negative inertial properties and refractive characteristics. Thus, periodic

structural configurations often form the basis for the design of metamaterials,

and photonic and phononic crystals [1, 2, 3, 4].

Several configurations of acoustic metamaterials and phononic crystals

have been proposed in recent years. These designs have illustrated the ability

to tailor equivalent material properties, and to achieve interesting dynamic

phenomena such as negative refraction [5, 6, 7, 8], wave beaming [9], and

bandgaps [10, 11, 12, 13]. Among the configurations proposed, the use of

plate structures as propagation media is an attractive solution due to the

existence of several theories to support the formulation of numerical models,

and the ease of experimentation provided by optical measurement systems

such as scanning laser vibrometers and interferometers. For these reasons, a

number of recent studies have investigated elastic plates with periodic inclu-

sions and voids, microstructures, and sources of impedance mismatch such

as surface mounted stubs [14, 15, 16] to generate phononic plates and elasto-

acoustic metamaterials [17, 18]. In particular, the work of [12, 8, 6, 7] focuses

on flexural (transverse) waves in periodically perforated plates.

2

The objective of this paper is to conduct the numerical and experimental

investigation of transverse wave motion in a periodically perforated plate.

Transversely polarized plate wave modes correspond to those modes charac-

terizing the low frequency/long wavelength Lamb wave spectrum and often

specifically to the first asymmetric mode (A0) [19]. At low enough frequen-

cies this mode can be well approximated by plate theories such as Kirchhoff’s

and Mindlin’s [20]. The configuration of the unit cell of the periodic assem-

bly consists of double-C perforations following Guenneau et al. [5], which

are also reminiscent of electromagnetic metamaterial designs [21]. This spe-

cific configuration is interesting for a number of reasons. First, the periodic

array of perforations is expected to provide the plate with partial and full

frequency bandgaps, i.e. frequency ranges within which wave propagation is

impeded either omnidirectionally or only for a specific range of directions.

Such bandgaps may be the result of Bragg scattering, or may be produced

by local resonances within the unit cell [22]. In addition, the non-isotropic

unit cell design is also expected to lead to an overall, frequency-dependent

anisotropy of the plate behavior, and consequently to the onset of wave di-

rectionality [9, 23].

A numerical model is developed for the perforated plate using rectangu-

lar Mindlin plate elements. The model is validated by comparing with the

predictions of a solid model. This also highlights the frequency range of accu-

racy of the formulation. Next, scanning laser vibrometer wave field measure-

ments are conducted to further validate the numerical model and highlight

the occurrence of anomalous wave motion, directional behavior and local res-

onances occurring within the considered plate. Specifically, wave field data

3

are recorded over the entire plate to compute dispersion relations through the

application of multidimensional (2D and 3D) Fourier transforms (FTs) [24],

to visualize the shape of propagating wavefronts, and compare them with

estimations based on dispersion surfaces and group velocity computations

based on unit cell analysis and the application of Bloch theorem [25].

The paper is organized in four sections, including this introduction. Sec-

tion 2 briefly describes the considered plate configuration, and presents the

formulation of the numerical model for the evaluation of the dispersion char-

acteristics of the plate. Section 3 illustrates the experimental setup, the

procedures, and the signal processing methodologies used for the evaluation

of the wave properties based on experimental data. Finally, Section 4 sum-

marizes the main findings of the work and provides recommendations for

future investigations.

2. Numerical analysis of wave motion

2.1. Configuration

The considered periodic plate and unit cell configuration are shown in

Fig. 1. This configuration is expected to be characterized by higher equiv-

alent stiffness along the longitudinal direction, and lower stiffness along the

vertical direction, and thus appears as inherently anisotropic. Stiffness di-

rectionality has previously been described as one essential mechanism for the

generation of frequency “caustics” [26, 27], which in turn are responsible for

the angular concentration of wave vectors at specific frequencies and their

directional interference. In addition, the double-C perforations naturally de-

fine a resonating unit within the unit cell, which leads to localized modes and

4

Ø10mm

25mm

25mm

Ø20mm

2mm

Figure 1: (a) Perforated plate, and (b) unit cell geometry and dimensions.

The thickness of the plate is 1 mm.

the generation of internal resonance frequency bandgaps [22, 28, 29]. The

considered plate arrangement therefore appears as a convenient configura-

tion for the numerical and experimental investigation of wave directionality

and resonant bandgaps, which are two of the predominant features of elastic

metamaterials.

2.2. Approach

A numerical model of the plate in Fig. 1 is developed to estimate its

dynamic behavior, and in particular to predict its dispersion properties and

corresponding wave velocities. The model is a finite element (FE) based

discretization of the structure by means of four-node Mindlin plate (MP)

elements [20]. MP elements are here used as a best compromise between the

ability to capture high frequency Lamb wave modes that are typically too

coarsely approximated by Kirchoff plate formulations, and the computational

cost that is associated with the three-dimensional (3D) discretization of the

5

plate. Remark, for at least the lower part of the frequency range considered

in this paper Kirchhoff plate formulation would have been just as good, but

the presented Mindlin formulation is directly applicable to higher frequencies

and thicker plates. Focus is placed on the out-of-plane motion (bending) of

the plate, which can be conveniently measured through the available scan-

ning laser vibrometer system. Results in the upcoming subsections illustrate

how the considered model is effective at capturing the low frequency predom-

inantly transversely polarized, or antisymmetric, Lamb modes of the plate,

which are of interest to the present study [19].

2.3. Governing equations

The out-of-plane dynamic behavior of the plate considered herein can be

described by the following coupled set of governing differential equations [30]:

∇TG (∇w −Θ) = ρh∂2w

∂t2(1)

∂TD∂Θ + G (∇w −Θ) =ρh3

12

∂2Θ

∂t2(2)

where w(x, t) denotes the out-of-plane deflection of a point of the mid-surface

at location x = [x, y]T , Θ = [θx, θy]T is a vector containing the the two

rotations of a material line normal to the mid-surface about the two in-plane

axes, ρ is the material density, while h is the plate thickness. Also, D = D(x)

and G = G(x) are the plate constitutive matrices. At a given point in the

plate material, which is locally isotropic, these are:

D =Eh3

12 (1− ν2)

1 ν 0

ν 1 0

0 0(1− ν)

2

(3)

6

and

G =E

2 (1 + ν)

5

6h 0

05

6h

(4)

Finally, the differential operator ∂ in Eq. (1) is defined as:

∂ =

1 0

0 0

0 1

∂

∂x+

0 0

0 1

1 0

∂

∂y= α1

∂

∂x+α2

∂

∂y(5)

The objective is to investigate the propagation of plane waves associated

with the bending motion of the periodic plate. Therefore, solutions in the

form of a Bloch-expansion [25]:w (x, t)

Θ (x, t)

=

w (x)

Θ (x)

ei(k·x+ωt) (6)

where w and Θ are periodic functions whose period coincides with the unit

cell, ω is the angular frequency, and k = [kx, ky]T is the wave vector.

Equation (1) can be rewritten in terms of the periodic fields to give:

∇Tk G

(∇kw − Θ

)= −ω2ρhw (7)

where a new operator ∇k is defined as

∇k = ∇ + ik = β1

(∂

∂x+ ikx

)+ β2

(∂

∂y+ iky

)(8)

where β1 = (1, 0)T , and β2 = (0, 1)T .

Similarly, Eq. (2) can be rewritten as

∂Tk D∂kΘ + G

(∇T

k w − Θ)

= −ω2ρh3

12Θ (9)

where the operator ∂k is defined as

∂k = α1

(∂

∂x+ ikx

)+α2

(∂

∂y+ iky

)(10)

7

2.4. Finite element discretization

Equations (7) and (9) can be discretized using a standard Galerkin ap-

proach. The result can be cast in the same form as the in-plane wave prop-

agation problem described in [31]. The mesh consists of 6560 square-shaped

Mindlin plate (MP) elements with side lengths of 0.25 mm. The mesh is

selected upon a convergence study conducted on the predicted dispersion re-

lations evaluated over the frequency range of interest in this study, as spec-

ified below. The smallest wavelength of interest is around half the unit cell

width, which means the chosen discretization should be fine enough to resolve

it. Also the mesh is refined enough to produce an acceptable description of

the internal geometry of the cell perforations, which cannot be perfectly re-

solved by the considered the mesh, as illustrated in Fig. 2. However, the use

of square-shaped elements simplifies the meshing procedure and the model

formulation as a whole, and represents a natural choice when considering a

possible future extension of this work to design optimization of the unit cell.

Furthermore, MP elements provide acceptable approximation of the lower

frequency wave modes of the plate with a computational cost that enables

the exploration of the full dispersion surfaces and the estimation of wave

velocities, which is one of the objectives of this study. In order to avoid

numerical inaccuracies due to shear locking, selective reduced integration of

shear deformation terms has been used when computing element stiffness

matrices.

2.5. Estimation of dispersion properties

The discretized equation for a unit cell can be generally expressed as:(K−ω2M

)u = 0 (11)

8

Figure 2: The mesh used to discretize the unit cell consists of 6560 square-

shaped elements.

where

K = K0 − i(kxK1 + kyK2) + k2xK3 + kxkyK4 + k2

yK5 (12)

and u is the vector of the generalized nodal displacements of the unit cell.

Expressions for the matrices above can be found in Appendix A. Solution of

Eq. (11) in terms of ω for an assigned wave vector k leads to the dispersion

relation of the periodic plate. As customary, the dispersion relations are

evaluated and plotted for a wave vector varying along the boundary of the

irreducible Brillouin zone, which leads to the band diagram representation.

The boundary of the irreducible Brillouin zone is the line O-A-B-C-O in Fig.

3.

The procedure above is applied to compute the band diagram for the

considered periodic domain, which is shown in Fig. 4. The diagram is esti-

mated by considering the mesh presented in Fig. 2, which corresponds to the

discretization of the thinnest member of the assembly being discretized by

eight elements, an even number that helps preserving the symmetry of the

geometry. This mesh provides an accurate estimation of the band diagram

9

Figure 3: Illustration of the first Brillouin zone and of the irreducible Bril-

louin zone (gray area). The parameters lx and ly are the unit cell’s width

and height, respectively.

within the considered frequency range. This is verified by comparing the cor-

responding diagrams with those obtained from a mesh with elements of half

the size. Such comparison does not show any noticeable difference, thus the

coarser mesh is used for follow on investigations because of computational

efficiency.

Figure 4 compares the dispersion branches estimated through the MP

elements (red dots), and through a 3D mesh consisting of eight-node solids

(open circles) using three elements to discretize the plate thickness. The 3D

model also provides the in-plane polarized modes (red circles), which mostly

appear independent from the predominantly out-of-plane branches (black

circles) in a frequency range up to approximately 40 kHz. The comparison

confirms that a plate model is suitable for wave propagation characterization

in the considered frequency range.

The contour plots of the first four dispersion surfaces corresponding to

10

O A B C O0

5

10

15

20

25

30

35

40

45

50

Fre

quen

cy [k

Hz]

Figure 4: Band diagram: comparison of MP model (red dots) and 3D model

predictions showing the decoupling of in-plane (red circles) and out-of-plane

(black circles) polarized modes in the considered range of frequencies.

the first four branches in Fig. 4 are shown in Fig. 5, which illustrates the

potential for directionality of the considered plate configuration. Namely,

the isofrequency contours for the first dispersion surface appear as approxi-

mately circular, which suggests a quasi-isotropic behavior in this frequency

range. In contrast, the dispersion surfaces for the third and fourth mode in

particular appear as mostly aligned along the vertical axis, at frequencies

around 7 and 9 kHz respectively. This suggests that directionality along the

horizontal direction may occur when wave motion at these frequencies and

according to these modes takes place within the plate. Directionality will

be further illustrated through the analysis of group velocities, based on the

developments presented in the next section.

11

kxlx/π

kyl y/π

−1 0 1−1

−0.5

0

0.5

1

1

2

3

4

(a)

kxlx/π

kyl y/π

−1 0 1−1

−0.5

0

0.5

1

3.5

4

4.5

5

5.5

(b)

kxlx/π

kyl y/π

−1 0 1−1

−0.5

0

0.5

1

6

6.5

7

7.5

(c)

kxlx/π

kyl y/π

−1 0 1−1

−0.5

0

0.5

1

7

8

9

10

11

12

(d)

Figure 5: Contour plots of dispersion surfaces for the first wave modes (con-

tour frequency values are expressed in kHz). (a) Mode 1. (b) Mode 2. (c)

Mode 3. (d) Mode 4.

Additional observations can be made upon the detailed analysis of the

out-of-plane branches of the dispersion diagram, which are shown in Fig. 6.

In this frequency range, the two largest partial bandgaps occur in the O-C

range with center frequencies of approximately 6.75 and 20 kHz. These fre-

quencies are highlighted by a dashed and solid line, respectively. The O-C

direction corresponds to the vertical direction in the plate, which suggests

that within these bands motion occurs predominantly along the horizontal

12

direction, thus indicating the directional behavior of wave motion. Further-

more, the branches bounding these partial bandgaps end with a zero slope at

O, which identifies a null group velocity. In particular, the lower bandgaps

appears to be bounded by branches that remain relatively flat through the

entire O-C wavenumber range, which characterizes wave motion remaining

localized as a result of the occurrence of internal resonant behavior. This is

illustrated by plotting the wave modes associated with these branches. The

local wave modes are plotted both at O, which corresponds to the classic

eigenmodes of the unit cell, and at the point half way in between O and C.

These mode shapes are all shown in Fig. 7, where it can be seen that the

mode changes slightly along the branch.

The localized nature of the modes associated with the lowest two branches

appears evident from Figs. 7 (a) and (b), which correspond to the modes

belonging to the branches starting at points a) and b) in Fig. 6 for frequencies

equal to 6.1 and 7.4 kHz. Both modes are characterized by the resonance

of the double-C inclusion. One is bending dominated and the other being

of a torsion type. The modes associated with the second partial bandgap

(Figs. 7 (c) and (d)) appear different in nature as the corresponding dynamic

deformations are no longer localized to the inclusion, but occur over the

entire unit cell. However, the out-of-phase motion of the internal inclusion

with respect to the surrounding medium appears as a common feature of all

the modes shown in Fig. 7.

2.6. Group velocity

The group velocity is a vectorial quantity that provides information re-

garding the speed and direction of wave propagation in the considered medium

13

O A B C O0

5

10

15

20

25

Fre

quen

cy [k

Hz]

a)b)

c)

d)

Figure 6: Illustration of partial bandgaps in the O-C range for out-of-plane

polarized branches. Center frequencies of approximately 6.75 and 20 kHz

are, respectively, highlighted by the black dashed and solid lines.

[25]. The evaluation of group velocities relies on the estimation of the disper-

sion relations, which can be conducted using the procedure illustrated in the

previous section. Subsequently, the estimation of the group velocity follows

the approach originally presented in [32] for the case of 1D photonic crystal.

The group velocity cg, defined as

cg = ∇ω =

[dω

dkx,

dω

dky

]T(13)

is estimated by considering the form of the eigenvalue problem in terms of ω

(Eq. (11)), i.e.:

G (k, ω(k),u(k)) =(K−ω2M

)u = 0 (14)

Differentiating with respect to k gives:

dG(k, ω(k),u(k))

dk=∂G

∂u

du

dk+∂G

∂ω

dω

dk+∂G

∂k(15)

14

(a) (b)

(c) (d)

Figure 7: Wave mode shapes associated with the branches bounding the

partial bandgaps. For each branch two modes have been plotted (left is at

O, right is halfway between O and C). Modes belonging to branch starting

at (a) 6.0 kHz. (b) 7.3 kHz. (c) 18.7 kHz. (d) 21.6 kHz.

The term du/dk can be eliminated by pre-multiplying by a vector vT :

vT ∂G

∂u

du

dk+ vT ∂G

∂ω

dω

dk+ vT ∂G

∂k= 0 (16)

and requiring that:

vT ∂G

∂u= 0 (17)

which corresponds to a left eigenvalue problem. Substituting Eq. (17) into

Eq. (15), the group velocity can be obtained as:

dω

dk= −

vT ∂G

∂k

vT∂G

∂ω

(18)

15

with

∂G

∂kx= (−iK1 + 2kxK3 + kyK4) u (19)

∂G

∂ky= (−iK2 + kxK4 + 2kyK5) u (20)

∂G

∂ω= −2ωMu (21)

In general, it is necessary to solve the eigenvalue problem in Eq. 17 to get

the left eigenvector v. However, if the presence of any dissipation mechanism

is neglected, both K and M are Hermitian and the left eigenvector v is the

complex conjugate of the right eigenvector, u.

In order to compute the group velocity at a given frequency ω, we need to

know the corresponding wave vector(s) k. Eq. (11) is a quadratic eigenvalue

problem in both kx and ky. However, by introducing an angle, θ, specifying

the direction of a given wave vector, the wave vector is given as:

k = [kx, ky]T = [k cos θ, k sin θ]T

such that Eq. (11) can be expressed as a standard eigenvalue problem in

terms of the wave vector magnitude k [33]:

(Kk(θ, ω)− kMk(θ)) uk= 0 (22)

Equation (22) can be solved in terms of the wave vector for an assigned

frequency ω and direction as defined by the angle θ.

Examples of group velocity plots computed at two selected frequencies

are shown in Figs. 8 (b) and (d), along with the corresponding isofrequency

contours of the dispersion surfaces (Figs. 8 (a) and (c)). The isofrequency

16

dispersion contour at 2 kHz appears as almost circular, which is an indication

of the quasi-isotropic behavior of the plate at low frequencies. The group ve-

locity directional plot at the same frequency shows that propagation appears

as occurring with a slightly higher velocity along the horizontal direction, as

shown by the lobar shape of the group velocity curve (Fig. 8 (b)). The case

of 11 kHz includes two sets of curve, as two wave modes contribute to prop-

agation at this frequency (Figs. 8 (c) and (d)). The group velocity variation

in Fig. 8 (d) shows two curves, one of which appears again elongated along

the horizontal direction, to indicate faster propagation along the horizontal

direction. The second curve is characterized by similar speeds along the ver-

tical direction and longitudinal one, however it appears to be characterized

by two sharper longitudinal lobes. The group velocity curves in Fig. 8 (b)

and (d) define the pattern of wave motion within the domain, and essentially

predict the shape of the propagating wavefront at the considered frequencies.

17

−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

kxlx/π

kyl y/π

(a)

−600 −400 −200 0 200 400 600−600

−400

−200

0

200

400

600

cgx [m/s]

cgx[m

/s]

(b)

−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

kxlx/π

kyl y/π

(c)

−600 −400 −200 0 200 400 600−600

−400

−200

0

200

400

600

cgx [m/s]

cgx[m

/s]

(d)

Figure 8: Isofrequency contour and corresponding group velocity profiles at

two selected frequencies. (a) Isofrequency contour 2 kHz. (b) Group velocity

2 kHz. (c) Isofrequency contour 11 kHz. (d) Group velocity 11 kHz.

18

3. Experimental investigation of wave motion

3.1. Setup

The wave propagation characteristics of the considered plate are inves-

tigated experimentally through measurements conducted using a scanning

laser vibrometer (SLV) (Polytec Model PSV 400 M2). The plate specimen

was fabricated out of an aluminum (Young’s Modulus E = 69 GPa, Poisson’s

ratio ν = 0.33, and the density ρ = 2700 kg/m3) plate of the dimensions con-

sidered for the numerical investigations. A 15×15 array of double-C perfora-

tions was obtained through water jet cutting. The resulting plate is shown in

Fig. 9, along with the detail of a unit cell. The center cell was not cut in order

to maintain a central location where a piezoelectric (PZT) disc is bonded to

apply the desired excitation. A PZT disc of 7 mm in diameter from Steminc

Corp. was selected as a source capable of introducing the desired excitations

at the frequencies/wavelengths of interest to the current study. The choice

is based on prior experience on guided wave fields generation and detection

in plate structures as described for example in [34]. The piezoelectric disc is

excited by a seven cycle Hanning modulated sine burst at the desired center

frequency. The plate’s out-of-plane response is recorded over a refined grid

of points prescribed through the SLV software. At each measurement point,

the recorded response is the result of ten averages taken to minimize distor-

tions due to measurement noise. Complete wave field images are obtained

by aggregating single-point measurements at each grid point. Measurements

and excitations are synchronized with a trigger signal so that proper phase is

maintained among all measurement points. Upon completing the scanning of

the measurement grid, the data are post-processed by the SLV software and

19

(a) (b)

Figure 9: (a) Experimental plate and (b) close-up containing detail of unit

cell.

coherent wave images are obtained. Examples of wave field images obtained

for two of the excitation frequencies considered are shown in Figs. 10 and

11.

3.2. Wave modes

A set of verification experiments is conducted to evaluate the modal char-

acteristics of the unit cell at some of the frequencies highlighted by the numer-

ical study. These experiments are conducted by considering a refined local

grid over a single unit cell. The plate is excited through harmonic excitation

at the frequencies of interest. Frequency domain acquisitions are conducted

to measure the dynamic deflection shapes of the unit cell, which are regarded

as closely related to the unit cell wave modes of the plate. Results at three

selected frequencies displayed in Fig. 11 show significant resemblance to the

wave modes presented in Fig. 7. This once again confirms the accuracy of

20

(a) (b) (c)

(d) (e) (f)

Figure 10: Snapshots of experimental wave fields for excitation at 3 kHz after

(a) t = 546µs, (b) t = 624µs, and (c) t = 702µs, and for excitation at 11 kHz

after (d) t = 312µs, (e) t = 390µs, and (f) t = 468µs.

the numerical implementation presented in the first part of this paper, and

its ability to predict local resonance characteristics at the unit cell level.

3.3. Evaluation of dispersion

The dispersion relations for the plate are estimated by processing the wave

field data recorded during the experiments. Wave field information can be

described as an array of matrices w(x, y, t) containing spatial and temporal

field information, where w denotes the measured out-of-plane displacement

21

(a) (b)

(c)

Figure 11: Dynamic deformed shapes at selected frequencies recorded

through local measurements at the unit cell level. (a) 6.0 kHz. (b) 7.3

kHz. (c) 19.3 kHz.

of the plate.

The dispersion properties are first estimated along the O-A and O-C di-

rections - in order to describe the relation between frequency and wavenum-

ber along two of the boundaries of the Brillouin zone - corresponding to the

horizontal and vertical directions, respectively. To this end, one of the coordi-

nates is kept fixed in the wave field matrix, and the two-dimensional Fourier

transform (2D-FT) in time and space (along the remaining coordinate and

time) is evaluated. This operation can be described as follows:

22

w(kx, ω) = F2D[w(x, y = y0, t)] (23)

w(ky, ω) = F2D[w(x = x0, y, t)] (24)

where F2D denotes the 2D-FT operator, w is the representation of the plate

response in the frequency/wavenumber (f − k) domain, while x0, y0 are the

coordinates along which dispersion is estimated.

The process is conducted by extracting data from the measurement grid

shown in Fig. 12, specifically along the horizontal and vertical line shown

in the figure, which correspond to the O-A and O-C directions of the band

diagram. The procedure is applied to data corresponding to various excita-

tion frequencies, such that a broader band of the spectrum can be covered

by superimposing the results of several narrowband experiments. Results

for the O-A and O-C directions are presented in Fig. 13, where the experi-

mental contours (from five experiments with excitation frequencies 2 kHz, 3

kHz, 8.5 kHz, 11 kHz, and 15 kHz) are overlaid to the numerically estimated

branches for the corresponding directions. As one can easily see, the exper-

imental branches appear in a wavenumber range that exceeds the limits of

the first Brillouin zone, as the spatial sampling considered includes several

measurement points within each unit cell. In order to enable a comparison

with numerical results, the numerical branches are therefore replicated pe-

riodically over subsequent intervals in multiples of π in order to observe or

resolve potential indetermination associated with branch folding, which is

the result of the investigation of Bloch wave modes.

Next, dispersion relations are visualized in the kx, ky domain through the

application of the 3D FT and the selection of specific frequencies of interest.

23

Figure 12: Experimental grid. Red lines indicate which measurement points

are used for evaluation of dispersion relation for O-A (horizontal line) and

O-C (vertical line).

This operation can be described as follows:

w(kx, ky, ω) = F3D[w(x, y, t)] (25)

where F3D denotes the 3D Fourier transform. Representation of w at a

specified frequency ω = ω0 leads to a surface w(kx, ky, ω0) whose magnitude

can be represented in the kx, ky domain, where direct comparison with the

dispersion surfaces at the selected frequency can be conducted.

The dispersion surfaces are evaluated through the Bloch analysis, as pre-

sented in the previous sections, conducted over a specified grid of the in-plane

wavenumber components. The isofrequency line of interest at ω = ω0 is ex-

tracted and overlaid to the experimental contour, as shown in Fig. 14. This

representation allows the evaluation of the energy content in the wavenum-

24

ber domain, and the evaluation of potential directionality induced by the

anisotropy of the plate at the selected frequencies. In Fig. 14, the numeri-

cal contours are black lines (Eq. 11) and dots (Eq. 22), while the surfaces

|w(kx, ky, ω0)|, each normalized to unity for convenience of representation, is

shown by the different contours, with red representing the highest values and

blue the lower values. For clarity, the boundaries of the first Brillouin zone

are also highlighted by the blue square included in the figures. As expected,

the numerical solutions found using Eq. 11 coincide with the solutions to Eq.

22. The experimental results confirm the previously observed fact that the

measured wavenumbers exceed the boundaries of the Brillouin zone, which

requires for the numerical dispersion to be computed beyond this range.

The numerical results show the expected periodicity in the wavenumber

domain, and generally overlay well with the experimental results. At low

frequencies, the results indicate that the plate behavior is approximately

isotropic, as demonstrated by the quasi-circular isofrequency lines obtained

experimentally and numerically (see Figs. 14 (a) and (b) for results corre-

sponding to 2 kHz and 3 kHz, respectively). At higher frequencies (see Figs.

14 (c) and (d) for results corresponding to 8.5 kHz and 11 kHz, respectively)

the distributions show occurrences of preferential directions corresponding

to peaks of the surface |w(kx, ky, ω0)| in the wavenumber domain. A good

agreement is observed, in particular for the low frequency cases. At higher

frequencies, the predictions appear less accurate which may be attributed

to the reduced fidelity of the model and to the contribution of other wave

modes not included in the numerical isofrequency contour displayed.

25

Fre

qu

ency

[kH

z]

kxlx/π0 0.5 1 1.5 2 2.5 3 3.5 4

0

5

10

15

(a)

kyly/π

Fre

qu

ency

[kH

z]

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

(b)

Figure 13: Dispersion relations k − f representation along the (a) O-A and

(b) O-C direction, and comparison with numerical predictions (red dotted

lines) (Numerical branches are replicated periodically in intervals of π).26

kxlx/π

kyl y/π

−2 0 2−3

−2

−1

0

1

2

3

(a)

kxlx/π

kyl y/π

−2 0 2−3

−2

−1

0

1

2

3

(b)

kxlx/π

kyl y/π

−2 0 2−3

−2

−1

0

1

2

3

(c)

kxlx/π

kyl y/π

−2 0 2−3

−2

−1

0

1

2

3

(d)

Figure 14: Experimental 3D FT (contour plots) and isofrequency dispersion

surfaces (black dotted line) at selected frequencies for representation of plate

dispersion in the kx − ky domain. (a) 2 kHz. (b) 3 kHz. (c) 8.5 kHz. (d)

11 kHz. Note that the numerical results are plotted both as contour slices

(solid black line), and as solutions to Eq. 22 (black dots) in which the angle

and frequency are specified before solving for k.

27

3.4. Directionality and group velocities

The dispersion plots presented in Fig. 14 reveal the complex plate be-

havior as the frequency increases, which is characterized by a transition from

quasi-isotropic motion, to a more complex pattern with some distinct prefer-

ential directions. This can be further investigated from the analysis of group

velocity at the selected frequencies and their representation in conjunction

with time snapshots of the wave field.

In theory, the directional group velocity plots of the kind shown in Fig.

8 describe the pattern of energy flow within the structure, and highlight the

concentration of wave vectors in preferential orientations. This is expected

to be reflected in the shape of the wavefront observed at specific time in-

stants. This representation is provided in Fig. 15, where the group velocity

is overlaid on the wave field generated by excitation at the four selected

frequency values. The group velocity and wave field snapshots confirm the

quasi-isotropic behaviors at 2 kHz, and show distinct directional characteris-

tics at the higher frequencies where wave motion occurs preferentially along

the horizontal direction at 8.5 kHz (see Fig. 15 (c)) and both the horizontal

and vertical direction as in the case of the wave field at 11 kHz shown in Fig.

15 (d).

In all cases presented, the group velocity predictions provide a good rep-

resentation of the wavefront and agree with the experimental measurements.

Such agreement is somewhat confounded by the fact the excitation does not

consist of a pure tone, but it is defined by a seven cycle burst, which therefore

includes a range of frequencies that partially excites other wave modes.

28

(a) (b)

(c) (d)

Figure 15: Experimental wave field and theoretical group velocities at the

corresponding excitation frequencies. (a) 2 kHz. (b) 3 kHz. (c) 8.5 kHz. (d)

11 kHz.

29

Further, quantitative insight into the directional behavior of the plate can

be gained by evaluating the envelope of a propagating waveform. Envelope

estimation is conducted through the application of the Hilbert transform

(HT) along a selected direction of propagation. The envelope evolution along

one of the coordinates, say the horizontal direction x, is estimated as:

E(x, t) = |H[w(x, y = y0, t)]| (26)

where H denotes the HT of the considered waveform.

The envelopes’ evolution at two frequency values along the x- and y-

directions are shown in Fig. 16 in the form of space-time amplitude maps

on a gray scale (with the black color being associated with the highest am-

plitude values). The plots also show propagation lines along the considered

directions obtained by considering the numerically computed group velocity

component along the considered directions. The lines, represented as red

dotted lines, are obtained from expressions of the kind of ±cgxt and ±cgyt.

These representations again confirm the accurate estimations of group ve-

locities obtained from the considered numerical model, which provides the

correct rate of envelope propagation. The plots in Fig. 16 clearly highlight

the different wave speeds for the considered mode of propagation along the

two principal directions of motion.

30

(a) (b)

(c) (d)

Figure 16: Space time evolution of wave motion along the vertical and hor-

izontal direction and rate of propagation predicted numerically through the

estimation of group velocity (red solid line). (a) 2 kHz, x-propagation. (b) 2

kHz, y-propagation. (c) 11 kHz, x-propagation. (d) 11 kHz, y-propagation.

31

4. Conclusion

The work presented the numerical and experimental investigation of wave

motion in a periodic plate with double-C perforations. The considered config-

uration was chosen because of its expected non-isotropic equivalent behavior,

which leads to directional wave motion. Furthermore, the periodic arrange-

ment of perforations produces partial bandgaps due to internal resonances.

These wave phenomena are first predicted through a numerical model formu-

lated using MP elements, and are then investigated experimentally through

the analysis of wave field images. Numerical and experimental results char-

acterize the 2D dispersion properties of the plate, and its group velocities as

a function of direction of propagation and frequency. Furthermore, detailed

modeling and measurements at the unit cell level identify modes of reso-

nance associated with partial bandgaps. The results presented in this study

illustrate the range of investigations that can be performed on periodic plate

waveguides and the wave properties that can be estimated through the pre-

sented numerical and experimental approaches. In addition, the considered

configuration can be used as a baseline for the design of planar waveguides

with desired directional and bandgap configurations through unit cell topol-

ogy optimization.

Acknowledgement

E. Andreassen’s support for this work was provided by the Danish Re-

search Agency through the innovation consortium F rMAT. M. Ruzzene and

K. Manktelow acknowledge the support from the National Science Founda-

tion under grants no. 0926776 and 1130368.

32

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37

Appendix A. System matrices

With the nodal displacements interpolated over the element using the

shape function matrices Nw and NΘ, as follows

ue =

Nw(x, y) 0

0 NΘ(x, y)

we

Θe

(A.1)

the corresponding expressions for the element matrices, in Eq. 11 are:

ke0 =

∫Ωe

−∇NTwG∇Nw

(NT

ΘG∇Nw

)TNT

ΘG∇Nw −∂NTΘD∂NΘ −NT

ΘGNΘ

dΩe

ke1 = −

∫Ωe

NTwβ

T1 G∇Nw −∇NT

wGβ1Nw −(NT

ΘGβ1Nw

)TNT

ΘGβ1Nw NTΘα

T1 D∂NΘ − ∂NT

ΘDα1NΘ

dΩe

ke2 = −

∫Ωe

NTwβ

T2 G∇Nw −∇NT

wGβ2Nw −(NT

ΘGβ2Nw

)TNT

ΘGβ2Nw NTΘα

T2 D∂NΘ − ∂NT

ΘDα2NΘ

dΩe

ke3 = −

∫Ωe

NTwβ

T1 Gβ1Nw 0

0 NTΘα

T1Dα1NΘ

dΩe

ke4 = −

∫Ωe

NTw (β1 + β2)

TG (β1 + β2)Nw 0

0 NTΘ (α1 +α2)

TD (α1 +α2)NΘ

dΩe

ke5 = −

∫Ωe

NTwβ

T2 Gβ2Nw 0

0 NTΘα

T2Dα2NΘ

dΩe

me = −∫

Ωe

ρhNTwNw 0

0ρh3

12NT

ΘNΘ

dΩe (A.2)

where Ωe is the domain of element e. The matrices in Eq. (11) and (12) are

found by summation of the element matrices, that is

K0 =n∑

e=1

ke0, K1 =

n∑e=1

ke1, etc (A.3)

where n is the total number of elements, and∑

should be understood as the

finite element assembly operator (see e. g. [20])

38